Clavius, Christoph, Geometria practica

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            <s xml:id="echoid-s9493" xml:space="preserve">
              <pb o="211" file="241" n="241" rhead="LIBER QVINTVS."/>
            tur; </s>
            <s xml:id="echoid-s9494" xml:space="preserve"> quod tam quadratum ex diametro quadrati deſcriptum duplum ſit
              <note symbol="a" position="right" xlink:label="note-241-01" xlink:href="note-241-01a" xml:space="preserve">ſchol. 47.
                <lb/>
              primi.</note>
            drati lateris, quam quadratum diametri ſphæræ quadratilateris Octaedri. </s>
            <s xml:id="echoid-s9495" xml:space="preserve">Se- miſsis verò huius diametri, altitudo erit vtriuſuis Pyramidis. </s>
            <s xml:id="echoid-s9496" xml:space="preserve">Quare ſi hęcalti-
              <lb/>
              <note symbol="b" position="right" xlink:label="note-241-02" xlink:href="note-241-02a" xml:space="preserve">14. tertii-
                <lb/>
              dec.</note>
            tudo ducatur in tertiam partem quadrati lateris, producetur area vnius pyrami-
              <lb/>
            dis, id eſt, ſemiſsis Octaedri: </s>
            <s xml:id="echoid-s9497" xml:space="preserve">ac proinde duplum huius pyramidis aream totius
              <lb/>
            Octaedri indicabit.</s>
            <s xml:id="echoid-s9498" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9499" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9500" xml:space="preserve">
              <emph style="sc">Deinde</emph>
            quia ductis ex centro Dodecaedri ad omnes eius angulos re-
              <lb/>
              <note position="right" xlink:label="note-241-03" xlink:href="note-241-03a" xml:space="preserve">Area Dode-
                <lb/>
              caedri.</note>
            ctis lineis, Dodecaedrum in 12. </s>
            <s xml:id="echoid-s9501" xml:space="preserve">pyramides pentagonas æquales diuiditur; </s>
            <s xml:id="echoid-s9502" xml:space="preserve">ſi area
              <lb/>
            vnius pyramidis per cap. </s>
            <s xml:id="echoid-s9503" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9504" xml:space="preserve">inuenta multip licetur per 12. </s>
            <s xml:id="echoid-s9505" xml:space="preserve">procreabitur area to-
              <lb/>
            tius Dodecaedri. </s>
            <s xml:id="echoid-s9506" xml:space="preserve">Vtautem vnius pyramidis area habeatur, neceſſe eſt, & </s>
            <s xml:id="echoid-s9507" xml:space="preserve">aream
              <lb/>
            baſis pentagonæ inueſtigare ex latere dato, per ea, quę lib. </s>
            <s xml:id="echoid-s9508" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9509" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9510" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9511" xml:space="preserve">ſcrip ſimus,
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            & </s>
            <s xml:id="echoid-s9512" xml:space="preserve">pyramidis altitu dinem, vtiam docebo. </s>
            <s xml:id="echoid-s9513" xml:space="preserve">Ex ſuperiori plano producto demit-
              <lb/>
            tatur ad planum baſis oppoſitæ linea perpendicularis: </s>
            <s xml:id="echoid-s9514" xml:space="preserve">Huius enim ſemiſsis di-
              <lb/>
            ligenter inquiſita in partibus lateris Dodecaedri, per inſtrumentum partium lib.
              <lb/>
            </s>
            <s xml:id="echoid-s9515" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9516" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9517" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9518" xml:space="preserve">conſtructum, dabit pyramidis altitudinem quęſitam, quemadmodum
              <lb/>
            & </s>
            <s xml:id="echoid-s9519" xml:space="preserve">tota perpendicularis altitudinem Dodecaedri exhibet. </s>
            <s xml:id="echoid-s9520" xml:space="preserve">Quamtamen Geo-
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            metricè ita quoq; </s>
            <s xml:id="echoid-s9521" xml:space="preserve">deprehendemus. </s>
            <s xml:id="echoid-s9522" xml:space="preserve"> Quia cubus in Dodecaedro deſcriptus
              <note symbol="c" position="right" xlink:label="note-241-04" xlink:href="note-241-04a" xml:space="preserve">8. quinti
                <lb/>
              dec.</note>
            dem ſphęra, qua Dodecaedrum, comprehenditur, eiuſquelatus vnum angulum
              <lb/>
            Pentagoni Dodecaedri ſubtendit; </s>
            <s xml:id="echoid-s9523" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s9524" xml:space="preserve">eadem diameter eſt ſphærę, Dodecae-
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            dri, & </s>
            <s xml:id="echoid-s9525" xml:space="preserve">cubi: </s>
            <s xml:id="echoid-s9526" xml:space="preserve"> Sirecta ſubtendens angulum pentagoni inueſtigetur,
              <note symbol="d" position="right" xlink:label="note-241-05" xlink:href="note-241-05a" xml:space="preserve">12. triang.
                <lb/>
              rectil.</note>
            latus cubi: </s>
            <s xml:id="echoid-s9527" xml:space="preserve"> Et quia diameter ſphærę potentia eſt tripla lateris cubi, ſi quadra- tumlateris cubi inuenti triplicetur, habebitur quadratum diametri ſphærę, vel
              <lb/>
              <note symbol="e" position="right" xlink:label="note-241-06" xlink:href="note-241-06a" xml:space="preserve">15. tertiidec.</note>
            cubi, cuius radix quadrata ipſam diametrum dabit. </s>
            <s xml:id="echoid-s9528" xml:space="preserve">Cum ergo diameter Dode-
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            caedri, & </s>
            <s xml:id="echoid-s9529" xml:space="preserve">altitudo eiuſdem centra baſium oppoſitarum coniungens ſe in centro
              <lb/>
            ſecentbifariam, venabimur ſemiſſem huius altitudinis, nimirum altitudinem py-
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            ramidis quęſitam, hacratione. </s>
            <s xml:id="echoid-s9530" xml:space="preserve">Concipiatur triangulum rectangulum, cuius ba-
              <lb/>
              <note position="right" xlink:label="note-241-07" xlink:href="note-241-07a" xml:space="preserve">Perpendicu-
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              laris è centro
                <lb/>
              ſphæræ ad ba-
                <lb/>
              ſem Dodecae-
                <lb/>
              dri.</note>
            ſis eſt ſemidiameter Dodecaedri nota, cum tota diameter proximè cognita ſit,
              <lb/>
            latera verò circa angulum rectum, altitudo pyramidis, & </s>
            <s xml:id="echoid-s9531" xml:space="preserve">ſemidiameter circuli
              <lb/>
            baſem Dodecaedri circumſcribentis. </s>
            <s xml:id="echoid-s9532" xml:space="preserve">Cum ergo ſemidiameter hæc cognoſci
              <lb/>
            poſsit, ex iis, quę lib. </s>
            <s xml:id="echoid-s9533" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9534" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9535" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9536" xml:space="preserve">docuimus, cognoſcetur quoque reliquum la- tus, altitudo videlicet pyramidis, quam quærimus. </s>
            <s xml:id="echoid-s9537" xml:space="preserve">Porrò ſemidiameter prædi-
              <lb/>
              <note symbol="f" position="right" xlink:label="note-241-08" xlink:href="note-241-08a" xml:space="preserve">3. triang.
                <lb/>
              rectil.</note>
            cti circuli pentagonum Dodecaed@i circumſcribentis ita quo quered detur no-
              <lb/>
            ta. </s>
            <s xml:id="echoid-s9538" xml:space="preserve">Quoniam latus pentagoni ſubtenditin
              <unsure/>
            eo circulo grad. </s>
            <s xml:id="echoid-s9539" xml:space="preserve">72. </s>
            <s xml:id="echoid-s9540" xml:space="preserve">& </s>
            <s xml:id="echoid-s9541" xml:space="preserve">latus Decago-
              <lb/>
              <note position="right" xlink:label="note-241-09" xlink:href="note-241-09a" xml:space="preserve">Semidiame-
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              ter circuls
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              pentagonum
                <lb/>
              Dodesaedri
                <lb/>
              circumſcri-
                <lb/>
              bentis.</note>
            nigrad. </s>
            <s xml:id="echoid-s9542" xml:space="preserve">36. </s>
            <s xml:id="echoid-s9543" xml:space="preserve">cognita erunt hæc latera in partibus ſinus totius. </s>
            <s xml:id="echoid-s9544" xml:space="preserve">Si ergo fiat, vt latus
              <lb/>
            pentagoni in partibus ſinus totius cognitum ad idem latus notum ex hypothe-
              <lb/>
            ſi, ita latus Decagoni in iiſdem partibus ſinus totius cogniti ad aliud, prodibit
              <lb/>
            Decagoni latus in menſura lateris pentagoni cognitum. </s>
            <s xml:id="echoid-s9545" xml:space="preserve"> Et quia latus penta- goni poteſt latera decagoni, & </s>
            <s xml:id="echoid-s9546" xml:space="preserve">Hexagoni eiuſdem circuli, ſi quadratum lateris
              <lb/>
            decagoni proximè cogniti detrahatur ex quadrato lateris pentagoni, reliquum
              <lb/>
              <note symbol="g" position="right" xlink:label="note-241-10" xlink:href="note-241-10a" xml:space="preserve">10. tertiidec.</note>
            fiet quadratum lateris Hexagoni, id eſt, ſemidiametri, ideo que eius radix qua-
              <lb/>
            drata ſemidiametrum exhibebit notam.</s>
            <s xml:id="echoid-s9547" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9548" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9549" xml:space="preserve">
              <emph style="sc">Postremo</emph>
            quia ductis ex centro Icoſaedri ad omnes eius angulos
              <lb/>
              <note position="right" xlink:label="note-241-11" xlink:href="note-241-11a" xml:space="preserve">Area Icoſae-
                <lb/>
              dri.</note>
            rectis lineis, Icoſaedrum in 20. </s>
            <s xml:id="echoid-s9550" xml:space="preserve">pyramides triangulares ęquales diuiditur; </s>
            <s xml:id="echoid-s9551" xml:space="preserve">ſi
              <lb/>
            area vnius pyramidis per cap. </s>
            <s xml:id="echoid-s9552" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9553" xml:space="preserve">inuenta multiplicetur per 20. </s>
            <s xml:id="echoid-s9554" xml:space="preserve">gignetur to-
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            tius Icoſaedri area ex illis 20. </s>
            <s xml:id="echoid-s9555" xml:space="preserve">pyramidibus conflata. </s>
            <s xml:id="echoid-s9556" xml:space="preserve">Vtautem vnius pyramidis
              <lb/>
            area obtineatur, inueſtiganda primum erit area baſis triangularis, ex iis, quę
              <lb/>
            lib. </s>
            <s xml:id="echoid-s9557" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9558" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s9559" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9560" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s9561" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9562" xml:space="preserve">& </s>
            <s xml:id="echoid-s9563" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9564" xml:space="preserve">ſcripſimus, Deinde altitudo pyramidis </s>
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